Flag-transitive automorphism groups of 2-designs with λ ≥ (r,λ)2 are not product type

In this note we show that a flag-transitive automorphism group G of a non-trivial 2-$(v,k,\lambda )$ design with $\lambda \ge {(r,\lambda )}^2$ is not of product action type. In conclusion, G is a primitive group of affine or almost simple type.